a non-stationary spatial weather generator for statistical

A non-stationary spatial weather generator for

statistical modelling of daily precipitation

Pradeebane VAITTINADA AYAR and Juliette BLANCHET

International Workshop on Stochastic Weather Generators for HydrologicalApplications

Berlin, Germany – 20th September 2017

Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives

Outline

1 Context

2 Overview of the model

3 Parameter estimation

4 First evaluations

5 Conclusions and Perspectives

P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro – Berlin, Sept. 20, 2017 2/18


Outline

1 Context



4 First evaluations




Context

SCHADEX : method for flood determination and dam dimensioning at EDF

Need for a realistic daily rainfall simulator to provide inputs for SCHADEX

Currently available : Regional MEWP [cf. Evin et al., 2016] univariate model

Aim to build a generic rainfall simulator (working over different catchements) :

accounting for spatial dependencies (daily time scale and one km2 spatialresolution)well performing for high precipitation quantiles at catchement scale

First choice : to model rain occurrence and intensity at thesame time only from observations.



(a) Ardeche catchement at Sauze

Ardèche at Sauze basin

Longitude (km)

Latit

ude

(km

)

48 212 376 541 705 869 1033 1198

1620

1771

1922

2073

2224

2375

2526

2677

(b) Stations locations and altitudes

0

500

1000

1500

2000

Elevation (m)

●

●●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

CHASSERADES

SAINTE−MARGUERITE−LAFIGERE

CUBIERES

LANGOGNE

BERZEME

SABLIERESALBA−LA−ROMAINE

BOURG−SAINT−ANDEOL

LOUBARESSEMIRABEL

PRIVAS

SAINT−ETIENNE−DE−LUGDARES

SAINT−MONTAN

BAGNOLS−LES−BAINS

CHATEAUNEUF−DE−RANDON

FLORAC

LE−PONT−DE−MONTVERT

SAINT−SAUVEUR−DE−GINESTOUX

VILLEFORT

LAPALUD

VALS−LES−BAINSCHAUDEYRAC

LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

CHASSERADES


CUBIERES

LANGOGNE

BERZEME



LOUBARESSEMIRABEL

PRIVAS


SAINT−MONTAN



FLORAC



VILLEFORT

LAPALUD


LE−BLEYMARD

Ardèche at Sauze

X (km) − Lambert II extended

Y (

km)

− L

ambe

rt II

ext

ende

d

●

●

●●

●

●●

●●

●●

●

●

●

●

●

●

●

●

ST−PIERREVILLE

MAYRESMONTPEZAT−SOUBEYROL

ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE

VALLON−PONT−D'ARC

MALONS

SENECHAS

ST−MAURICE−DE−VENTALON

MAS−DE−LA−BARQUE

BESSEGES

STE−EULALIE

USCALADES−ET−RIEUTORD

MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC

MASMEJEAN/−BASTIDE−PUYLAURENT

ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


ST−PIERREVILLE


ANTRAIGUES−AIZAC

AUBENAS

LABLACHÈRE

JOYEUSE


MALONS

SENECHAS



BESSEGES

STE−EULALIE


MAZAN

LAC−D'ISSARLÈS

ISSANLAS−MEZEYRAC


695 709 722 736 749 763 776 790

1910

1921

1933

1944

1956

1967

1979

1990

FIGURE 1: Area : 2260 km2 – 42 Stations – Altitudes : from 47 to 1425 mAt least 20 years long time series



Outline

1 Context



4 First evaluations




Model framework

∀x ∈ D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}x∈D

What is {Y (x)}x∈D ?The rainfield {Y (x)}x∈D is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].

Gaussian field : fully defined by its mean vector and covariance structure.

Model stepsMarginal model

Spatial model



Model framework

∀x ∈ D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}x∈D

What is {Y (x)}x∈D ?The rainfield {Y (x)}x∈D is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].Gaussian field : fully defined by its mean vector and covariance structure.

Model stepsMarginal model

Spatial model



Marginal model

Extended Generalized Pareto of Type II (EGPII)Ü Model the full range of rainfall intensities without any threshold selection.Naveau et al. [2016] : P(Y 6 y) = G

(Hξ( yσ

))

where

Hξ, the Generalized Pareto Distribution

G(v) = vκ

Courtesy : Naveau et al. [2016] ξ = 0.5

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5

κ = 1κ = 2κ = 5Gamma

0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))

prec

ipita

tion

(mm

/day

)


ξ=0.510 20 50 100 1000 10000

Temps de retour T (annees)

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5


0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))

prec

ipita

tion

(mm

/day

)


ξ=0.510 20 50 100 1000 10000




Marginal model


(Hξ( yσ

))where


G(v) = vκ


0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5


0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))

prec

ipita

tion

(mm

/day

)


ξ=0.510 20 50 100 1000 10000


0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5


0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))pr

ecip

itatio

n (m

m/d

ay)


ξ=0.510 20 50 100 1000 10000




Marginal model


(Hξ( yσ

))where


G(v) = vκ

Constraints

Low values driven by κ, large values driven by ξ

Weilbull type lower tail behavior (bounded orshort-tailed)

Frechet type upper tail behavior (unbouded orheavy-tailed)


0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5


0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))

prec

ipita

tion

(mm

/day

)


ξ=0.510 20 50 100 1000 10000


0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

x

Den

sity

f(x)

ξ=0.5


0 2 4 6 8 10

050

100

150

200

U = −ln(−ln(F))pr

ecip

itatio

n (m

m/d

ay)


ξ=0.510 20 50 100 1000 10000




Marginal model

Extended GP-II : without threshold

F (y ;σ, ξ) =

(

1 −(

1 + ξyσ

)−1/ξ)κ

if ξ 6= 0,(1 − exp

{−yσ

})κ, if ξ = 0.

ξ is set to 0 Þ powered exponential used as marginaldistribution.



Marginal model

Extended GP-II : without threshold

F (y ;σ, ξ) =

(

1 −(

1 + ξyσ

)−1/ξ)κ

if ξ 6= 0,(1 − exp

{−yσ

})κ, if ξ = 0.

ξ > 0 Ü too large return levels for long return periods e.g. 1000 to 10000 years(standard return period in dam construction)

ξ is set to 0 Þ powered exponential used as marginaldistribution.



Spatial dependance

Let the following transformation :H(x) = Φ−1(F (Y (x))), where Φ is the CDF of the normal N(0, 1).

Þ Since Y (x) > 0, H(x) > T (x) ≡ Φ−1(FY(x)(0))

Þ Then H(x) = max(T (x),G(x)),

G(x) the latent Gaussian process GP(0, 1,C(h))with marginal N(0, 1), C(h) the spatial covariance function.

ProcedureEstimate C(h) from the H-transformed data,

Simulate G and deduce H,

Retrieve Y by transforming back with Y (x) = F−1Y(x)(Φ(H(x))).



Covariance function

CS(h) = σ2exp(−hλ

), the stationary exponential covariance (isotropic) function

Geometrical anisotropy widely used in the literature

h(xi , xj) =√

(xi − xj)TΣ−1(xi − xj)

with Σ a covariance matrix, Σ−1 = MT M :

M =

(cosψ sinψ

−b sinψ b cosψ

)

with b the elongation coefficient, ψ ∈[−π2 ,

π2

]



Covariance function




h(xi , xj) =√



M =

(cosψ sinψ

−b sinψ b cosψ

)


π2

]How to include topographical information in the

covariance ?



Covariance function




h(xi , xj) =√



M =

(cosψ sinψ

−b sinψ b cosψ

)


π2

]Non-stationarity introduced by Higdon et al. [1999] : kernel

convolution in anisotropic function.



Spatial non-stationary covariance function

Generalized by Paciorek and Schervish [2006]

One solution for spatial non-stationarity :

CNS(xi , xj) = |Σi |14 |Σj |

14

∣∣∣∣Σi + Σj

2

∣∣∣∣− 12

exp

−

√(xi − xj)T

(Σi + Σj

2

)−1

(xi − xj)

Σi ,Σj : 2× 2 covariance matrix (local kernels, ∀i, j Σi = Σj ) Ü classical anisotropy∀xi ∈ D, Σ−1

i = MTi Mi :

Mi =1λi

(cosψi sinψi

−bi sinψi bi cosψi

)=

(Ai cosψi Ai sinψi

−Bi sinψi Bi cosψi

)

with Ai , Bi (> 0) et ψi as functions of the covariates (lon, lat, elev).






CNS(xi , xj) = |Σi |14 |Σj |

14

∣∣∣∣Σi + Σj

2

∣∣∣∣− 12

exp

−

√(xi − xj)T

(Σi + Σj

2

)−1

(xi − xj)

Σi ,Σj : 2× 2 covariance matrix (local kernels, ∀i, j Σi = Σj ) Ü classical anisotropy

∀xi ∈ D, Σ−1i = MT

i Mi :

Mi =1λi

(cosψi sinψi


)=



)







CNS(xi , xj) = |Σi |14 |Σj |

14

∣∣∣∣Σi + Σj

2

∣∣∣∣− 12

exp

−

√(xi − xj)T

(Σi + Σj

2

)−1

(xi − xj)

Σi ,Σj : 2× 2 covariance matrix (local kernels, ∀i, j Σi = Σj ) Ü classical anisotropy∀xi ∈ D, Σ−1

i = MTi Mi :

Mi =1λi

(cosψi sinψi


)=



)




Outline

1 Context



4 First evaluations




The estimation procedure is performed over six sub-set based on :

a rainy days classification in K = 3 weather types [WT,Garavaglia et al., 2010],

S = 2 seasons defined as low risk (LO dec. to aug.) and high risk (HI sep. to nov.)seasons.

Marginal distribution parameters : maximum likelihood,

Covariance function parameters : censored maximum likelihood [Pesonenet al., 2015].



Covariance estimation and censored likelihood

gd(xi) not observed : only gd(xi) > Td(xi).

Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),

Let Od = {xi |gd(xi)is observed}

β the parameters of C(h)

The censored likelihood given the {gd(xi)}xi∈Zd is Lc(β) =∏

d

Lcd(β) with :

1 Lcd(β) = fGP(gd(x1), . . . , gd(xN);β), if it rains at all stations

2 Lcd(β) = fGP({gd(xi)}xi∈Od ;β)

×P({Gd(xi) 6 Td(xi)}xi∈Zd |{Gd(xi) = gd(xi)}xi∈Od ;β),if not

fGP are GP densities with covariance function C

P({Gd(xi) 6 Td(xi)}xi∈Zd |{Gd(xi) = gd ,m(xi)}xi∈Od ;β) is a conditional GP CDF



Outline

1 Context



4 First evaluations




First evaluations

Calibration over the whole period with available data (1948-2013),

100 simulations over the period,

The simulations are only conditioned by the season and weather type the daybelongs to.

Evaluation of the marginal and spatial dependance properties. In particular, theadded-value of NS-covariance is explored. To this end three different types ofexponential covariance function are tested :

isotropic (ISO)anisotropic (AN)non-stationary (NSAN)



Mean rainfall & occurrence frequency (LO WT2)

●●

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●

● ●●

●●

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●

●●

●

●●● ●

●

● ●

●●

● ●

●

●

4 6 8 10 12

Mean

NSAN● ●

●

●

●●●

●

●

●●

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●

● ●

●

●●

● ●●

10 15 20 25 30

Mean>0

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●

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● ●

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●

●●

30 35 40 45 50 55

P1

0

200

400

600

800

1000

1200

1400

Elevation (m)

●

●

●●

● ●

●●●

● ●

●

4 6 8 10 12

AN●● ●

●

●

●

●

● ●

●

●

●●

10 15 20 25 30

●●

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● ●●

●

●

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●

● ●●●● ●

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● ●

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● ●●

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●

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●●

●

30 35 40 45 50 55

●●●●

●

●

●●

●

●

●

4 6 8 10 12

ISOPR (mm/day)

●

●

●

●

●

● ●

●

● ●●

●● ●

● ●

●

●●●

●●

●

10 15 20 25 30

PR (mm/day)●

●

●

● ●

●● ●

●

●

●

●●

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●

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● ●

●

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●

●

●

● ● ●●●

●

30 35 40 45 50 55

%



Interannual variabilityLO WT2

020

040

060

080

010

00

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: NSAN

MeanQ5−Q95OBS

RMSE = 146.66 mmCOR = 0.49

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 147.44 mmCOR = 0.47

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 146.08 mmCOR = 0.48

020

040

060

080

010

00

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010


MeanQ5−Q95OBS

RMSE = 146.66 mmCOR = 0.49

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 147.44 mmCOR = 0.47

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 146.08 mmCOR = 0.48

020

040

060

080

010

00

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010


MeanQ5−Q95OBS

RMSE = 146.66 mmCOR = 0.49

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 147.44 mmCOR = 0.47

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 146.08 mmCOR = 0.48

NSAN

AN

ISO

HI WT2

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: NSAN

MeanQ5−Q95OBS

RMSE = 138.21 mmCOR = 0.45

020

040

060

080

0Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 137.45 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 137.51 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010


MeanQ5−Q95OBS

RMSE = 138.21 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 137.45 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 137.51 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010


MeanQ5−Q95OBS

RMSE = 138.21 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: AN

MeanQ5−Q95OBS

RMSE = 137.45 mmCOR = 0.45

020

040

060

080

0

Years

PR

(m

m/y

ear)

1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

RMSE = 137.51 mmCOR = 0.45



Log-odds ratio

The log-odds ratio can be seen as analogous to spatial correlation of continuousvariable for binary variables [Charles et al., 1999]

TABLE 1: Contingency table for two stations xi, xj

NR(xi) R(xi)

NR(xj) n00 n01

R(xj) n10 n11

The log-odds ratio is given by :

LOR = log(

n00n11

n01n10

)



Log-odds ratioLO WT2

−

−

−

−−

− −

−

−−

−

−

−−−−−

−

−−−−

−−

−−−−−−−−−−−−− −−−

−

−−−

−

− −

−

−−−

−−− −

−

−

−

−−−

−−−−−−−−

−

−− −−−−−−−−

−−−

−− −

−−−

−

−

−−−

−

−−

−−

−

−

−−−− − −− −−−−−

− −−−−−−−

− −

−−−

−

−

−−−− −−

−−

−−

−−−−−−−−

−−−−−

−− −−−

−−

−−

−

−−

−−

−−− −−− −

−

−−

−−−−−

−−−−−−−−−− −

−

−−−

−−−−−

−

−

−−

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HI GT2 cov mod: ISO


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HI GT2 cov mod: AN


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Correlogram (zero included)LO WT2

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.1GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.11GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.09GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.1GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.11GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.09GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.1GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.11GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

LO GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.09GT2 COR = 0.99

NSAN

AN

ISO

HI WT2

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88


MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.98

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: AN

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99

0.0

0.2

0.4

0.6

0.8

1.0

Distance (km)

Cor

rela

tion

0 6 12 19 27 35 42 50 58 65 73 81 88

HI GT2 cov mod: ISO

MeanQ5−Q95OBS

GT2 RMSE = 0.04GT2 COR = 0.99



Outline

1 Context



4 First evaluations




Conclusions & Perspectives

Good agreement in terms of marginal statistics with an underestimation of therain occurrence probability at stations

Underestimation of the spatial dependency

NS-covariance does not really improve the results in terms of marginal statisticsand spatial dependency (Is it worth it ?)




Good agreement in terms of marginal statistics with an underestimation of therain occurrence probability at stations

Underestimation of the spatial dependency

NS-covariance does not really improve the results in terms of marginal statisticsand spatial dependency (Is it worth it ?)

Another way to characterize spatial dependency,

Add cross-validation procedure,

Remove days with too many missing values in the dataset,

3-days cumulated amounts are really important for flood determination : Þadd temporal correlation (AR process)




LO WT2

HI WT2


Thank you for yourattention

Reference

Allard, Denis and Bourotte, Marc [2015]. “Disaggregating daily precipitations into hourly values with a transformed censored latentGaussian process”. In: Stochastic Environmental Research and Risk Assessment. Vol. 29. no. 2, pp. 453–462.

Charles, Stephen P., Bates, Bryson C., and Hughes, James P. [1999]. “A spatiotemporal model for downscaling precipitationoccurrence and amounts”. In: Journal of Geophysical Research : Atmospheres. Vol. 104. no. D24, pp. 31657–31669.

Evin, G., Blanchet, J., Paquet, E., Garavaglia, F., and Penot, D. [2016]. “A regional model for extreme rainfall based on weatherpatterns subsampling”. In: Journal of Hydrology. Vol. 541, Part B, pp. 1185 –1198.

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